How to check universality of quantum gates?

Abstract

We provide two simple universality criteria. Our first criterion states that $\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms a $\delta$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\ge 3$. Our second universality criterion says that $\mathcal{S}\subset G_d$ is universal if and only if the centralizer of ${\mathcal{S}}^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes {\bar{U}}^{\otimes t(d)}|U\in \mathcal{S}\}$ is equal to the centralizer of $G_d^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes {\bar{U}}^{\otimes t(d)}|U\in G_d\}$, where $t(2)=3$, and $t(d)=2$ for $d\ge 3$. The equality of the centralizers can be verified by comparing their dimensions.

arXiv comment5 pages, quadratically improved centralizer condition for universality, some typos fixed and a discussion of the results added. The title of published version was changed to "Universality verification for a set of quantum gates"

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Cited by in corpus (4)

• Virtual distillation with noise dilution
• Linear programming with unitary-equivariant constraints
• Optimized numerical gradient and Hessian estimation for variational quantum algorithms
• Matrix concentration inequalities and efficiency of random universal sets of quantum gates